1

Retro-analytical Reasoning IQ tests for the High

Range.

Marco Ripà

1 and Gaetano Morelli

2

1 sPIqr Society Founder, Rome, Italy

e-mail: marcokrt1984@yahoo.it

2 STHIQ Society Founder, Caserta, Italy

e-mail: tany.morelli@gmail.com

Abstract

In this paper we provide a proposal to produce a new, computer-based generation of IQ tests for the

high range (from +3σ and higher above the mean) without being prone to cheating.

IQ tests are at a considerable risk of this kind, and episodes of cheating have actually occurred, even

on the most famous standardized and supervised IQ tests.

As soon as a test is administered to a candidate, there is no longer the certainty that its items will

remain secret; this is as a consequence of the “static nature” of IQ tests, and the problem may be

solved through a dynamic system based on different items for each submission of the test.

Our proposal is based on a method to construct new integer sequences starting from a given and

explicit set of sequences, using information asymmetry (i.e. different items for each session). The

related solving of problems will be linked to inference and retro-analytical reasoning, similar to the

retrograde analysis of chess problems.

Keywords

IQ test, retro-analytics reasoning, World Intelligence Network, gifted, giftedness screening,

derivation process, standardized tests, cheating, integer sequences, Automatic Sequences Generator,

High Range Test, Italy.

2

1 Introduction and problem statement: IQ 200+ (σ=24) for the

average candidate

Most of the standardized and supervised IQ tests are not without risk if we want to use them for

giftedness screening (even if they represent the best choice for an average person’s reasoning skills

evaluation, or IQ deficit diagnosis).

It is regrettable that certain people try to sell (usually on the Web) some excellent standardized

supervised tests, as evidenced, for example, by the following screenshots (taken from eBay):

Figure 1: The result of a professional search of IQ tests on eBay using the names of some famous standardized and

supervised tests as keywords.

3

If someone succeeded in buying some tests of this kind, it would be easy to cheat on them,

achieving a perfect score under the supervision of a serious psychologist, in front of the Media too.

A couple of years ago, a displeasing episode occurred featuring the Cattell Culture Fair III (form

A+B) done by a candidate who got a perfect score under the television eye, but who was unable to

reach a 130 (σ=15) performance on a similar test, i.e. Raven’s Advanced Progressive Matrices with

a time limit of 60 minutes [11].

In order to reduce this cheating risk, one possible solution is to adopt qualitative [8] IQ tests, even if

episodes of cheating have also occurred on them (for example, the Get-y test attack in 2010,

performed by a group of Chinese [http://www.epiqsociety.net/get/]).

As far as unsupervised IQ tests for the high range are concerned, there are many examples of

“excellent works” which, unfortunately, are no longer accepted for admission to high IQ societies

[2].

The following text is taken from the official membership page of the “Prometheus Society”

[http://216.224.180.96/~prom/oldsite/membership/index.html]:

“The Prometheus Society officers have voted that Mega27 score sheets dated after 11/27/99 will

not be accepted for admission to the Prometheus Society. This measure is being taken in

response to recent compromises of the Mega27 test.”

The following text comes from the official membership page of the “One-in-a-Thousand Society”

(OATH) [http://www.oathsociety.com/membership.html]:

SOME QUALIFYING SCORES:

……………………………………

Mega Test (before 1995) 24 right

Titan Test 24 right (not taken after 2/27/2011)

Langdon Adult Intelligence Test (before 1994) 150 IQ

……………………………………

Besides, on the “Artistic Minds Society” [http://www.materials-synthesis.com/amc/], you can read:

“The CFNSE test has unfortunately been compromised and it is no longer being scored. Older

CFSNE scores though will still be accepted.”

Several other examples of IQ test scores, which are accepted in most high IQ societies “only if got

before certain dates”, can be found quite easily, simply by looking at the membership pages of these

groups.

4

The most common reason why some unsupervised high range IQ tests are no longer accepted is

related to their impairment, often due to the publication of the solution to some items on the Web,

or within other public contexts.

Of course, great tests, like the ones previously mentioned, are still useful (if still scored by the

authors) for auto-evaluation purposes. Nevertheless, each time a test is no longer accepted for

admission into certain high IQ societies, inevitably, it loses part of its interest.

With reference to what we have explained above, it would be optimal to discover at least one family

of supervised IQ tests for the high range which is immune from the risk of cheating. It is important

to point out that the risk we are talking about involves, almost exclusively, tests used with the aim

of investigating high performances.

A new proposal to create a new kind of totally culture free numerical high IQ test is based on a

method to construct new integer sequences starting from a given and explicit set of sequences. The

related solving of problems will be linked to inference and retro-analytics reasoning, similar to the

retrograde analysis of chess problems [7-9]. Our idea is to provide some software which is able to

produce a huge amount of “varying difficulty” items, starting from a short and public input. This

method is quite similar to the RSA encryption algorithm and is based on asymmetric information.

The candidates have to provide an articulated answer for each item and this approach cuts out for

the most part the “false positive results”.

2.1 The derivation sequences method to create a wide set of

supervised high range IQ tests

The method we are going to describe assures us about the possibility of creating a wide set of

distinct IQ tests, with an equivalent raw scoresIQ scores distribution. This means that, when the

norming process has been completed, the raw score obtained in any given test belonging to this

family will be associated with only one IQ score, regardless of the specific test taken. Passing from

one test to another one, the raw scoreIQ score conversion table will remain unchanged: this will

provide us with the possibility of using a given set of items “in only one session”, which is, as we

explained above, the main target of this work.

For the norming purpose, we can use one (random) specific test, or a small set of tests. We need to

calculate different conversion tables (which implies the same number of norms) if we want to test

children below 17 years of age; furthermore, if we are interested in IQ tests of various time

durations, or if we are focusing on different IQs ranges, the nearer the result is to the middle of the

range we are considering, the smaller the relative statistical error will be [3].

5

2.2 Main idea

Our method is based on a process to derive new sets of sequences starting from a small amount of

fixed sequences (9 to 12). The underlying rules are public. They are included in the incipit of the

test administered by the psychologist/psychometrist.

In order to give the candidates the wherewithal for understanding how each item is constructed, we

need to point out to them the mathematical idea which is at the bottom of the “derivation process”

we have adopted.

The Description of the “derivation process.”

The derivation process is the process we use to create an item of the final IQ test, starting from a

given set of integer sequences. The whole process consists of one or more “steps of derivation”

(that is, some items are created through a single step of derivation, other items through more than

one step).

Each step involves two sequences:

The first one (from here on, “FS”) represents a “sequence of values”, because some of its

elements are selected (according to a certain criterion which we are going to describe) and

used as “values of the elements which constitute the output sequence” (from here on, “OS”),

which is the sequence produced by the current step of derivation.

The FS is included in the initial set of integer sequences for the first step of derivation (in

this case, we call the FS the “mother sequence”); for steps of derivation whose order is

higher than one, the FS is the OS of the previous step.

The second sequence (from here on, “SS”) represents a “sequence of positions”, because the

various values of its elements indicate “the positions occupied by the elements of the FS

which will be inserted into the OS”.

During a given step of derivation, we run through the SS. For each element of the SS, we insert into

the OS an element belonging to the FS: the element chosen from the FS is the one which occupies

(within the FS itself) a position equal to the value of the current element of the SS. According to

this methodology, the SS represents the “rule” that is applied to the FS, in order to identify which

elements of the latter will be inserted into the OS of the current step of derivation.

The following table shows an example of the derivation process; both the mother sequence and the

SS constitute the canonical prime sequence (OEIS A000040 - http://oeis.org/A000040):

Zero order

primes, p<100

First order

primes, p<100

Second order

primes, p<100

Third order

primes, p<100

Fourth order

primes, p<100

2 ///

3 3 ///

5 5 5 ///

6

7

11 11 11 11 ///

13

17 17

19

23

29

31 31 31 31 31

37

41 41

43

47

53

59 59 59

61

67 67

71

73

79

83 83

89

97

Table 1: The derivation process considering as unique rule of derivation the Prime numbers < 100 subset.

Using the same principle above, we can define the set of the derived sequences which are

constructed from the rule “the cube of a conventional prime”. The smallest of them is 19, the eighth

conventional prime: in fact, 8=23.

What has been shown for the positional primes is valid for any integer sequence: it is just sufficient

to apply the same criterion iteratively.

For example (putting between brackets the progressive positional numbers referring to the next term

of the sequence):

Fibonacci sequence zero order (mother sequence) ↔

↔ (1)1,(2)1,(3)2,(4)3,(5)5,(6)8,(7)13,(8)21,(9)34,(10)55,(11)89,(12)144,(13)233,…

Order 1 sequence ↔ (1)1,(2)1,(3)1,(4)2,(5)5,(6)21,(7)233,(8)10946,(9)5702887,…

Order 2 sequence ↔ (1)1,(2)1,(3)1,(4)1,(5)5,…

…

Order 4 sequence n≥2 ↔ (1)1,(2)1,(3)1,(4)1,(5)5,…

Observation. In this case, when the order of the derived sequence is ≥2, the first 5 terms are fixed:

(1)1,(2)1,(3)1,(4)1,(5)5 (for further investigations, see the appendix below).

7

Combining multiple rules (related to different mother sequences), we can create an unlimited

number of order≥1 subsequences.

For example:

Mother sequence (zero order): Fibonacci

[(1)1,(2)1,(3)2,(4)3,(5)5,(6)8,(7)13,(8)21,(9)34,(10)55,(11)89,(12)144,(13)233,(14)377,(15)610,

(16)987,…]

Order 1 sequence: Fibonacci

[(1)1,(2)1,(3)2,(4)3,(5)5,(6)8,(7)13,(8)21,(9)34,(10)55,(11)89,(12)144,(13)233,(14)377,(15)610,

(16)987,…]

Order 2 sequence: Prime numbers

[(1)2,(2)3,(3)5,(4)7,(5)11,(6)13,(7)17,(8)19,(9)23,(10)29,(11)31,(12)37,(13)41,(14)43,(15)47,

(16)53,…]

Order 3 sequence: Perfect squares

[(1)1,(2)4,(3)9,(4)16,(5)25,(6)36,(7)49,(8)64,(9)81,(10)100,(11)121,(12)144,…]

Evolution:

Order 1 sequence (Fibonacci Fibonacci) ↔

↔ (1)1,(2)1,(3)1,(4)2,(5)5,(6)21,(7)233,(8)10946,(9)5702887,(10)139583862445,…

Order 2 sequence (Fibonacci Fibonacci Prime numbers) ↔ (1)1,(2)1,(3)5,(4)233,…

Order 3 sequence (Fibonacci Fibonacci Prime numbers Perfect squares) ↔ (1)1,(2)233,…

Observation. As already described above, it is self-evident that the second and the thirteenth

element (say a2 and a13) from the “mother sequence” also belong to the “order 3 sequence” and, for

the same reason, they fit all the order ≤3 sequences as well, according to the established hierarchical

structure.

Starting from a given sequence, it is possible to create infinite others (also coincident), but the

condition (necessary, but not sufficient) postulates that the set of the terms that compose it has

infinite cardinality (that is, the number of elements of the mother sequence must be infinite).

We have to distinguish between two kinds of sequences:

incremental sequences: sequences such that an<a(n+1), where an indicates the n-th term of the

sequence;

non incremental sequences: sequences which do not respect the previous property.

In the second case, proceeding with the derived sequences (order greater than 1), we could obtain an

endless chain of sequences which, starting from a certain step, repeat themselves with a precise

regularity (infinitely). This happens when the original sequence admits an upper bound. Anyway

8

this is a necessary, but not sufficient, condition (just think of an odd sequence whose terms are

always equal to 3, while the even elements are constituted by powers of 2, arranged in ascending

order: starting from the second step, the sequence will admit 3 as a majorant). Let us think, as

“borderline cases”, of the sequences derived from the decimal expansions of “e”, “pi” and “√ ”

(see Appendix), whose terms are strictly less than 10 (in these cases, you can choose m≥10 as a

majorant).

N.B.

For practical use, we denote by “IS” the mother sequences used by the algorithm (derivation

process) and “OS” for the final outputs (the sequences obtained considering the amount of “steps”

we want).

3 Concrete rules for the implementation of the test

These are the operating rules to construct one IQ test item:

1- Maximum amount of derivations: 3 steps (which means 3 underlying rules).

2- Not more than 15% of items based on 4 different rules.

3- Globally, each test should contain between 25 and 40 items.

4- Every term must be smaller than 1010

(from 1 to 9 digit long numbers).

5- Each IS should contain a minimum of 500 terms, while the OS must contain from 7 up to a

maximum of 20 elements (the text will show the minimum quantity of terms which is sufficient to

get a unique solution to each given item – see rule 10).

6- Items have to be randomly shuffled inside the test (with no hints about the difficulty of the item

itself, and without items sorted by difficulty).

7- The amount of items based on a given number of derivations (for example, 3 with only one

derivation, 20 with 2 derivations, 12 with 3 derivations) must be public and will be included inside

the pamphlet delivered to each candidate during the preliminary briefing (see section 6).

8- The text (public sequences set) has to contain between 50% and 65% of strictly non decremental

sequences (each term must be smaller than the next one) and from 35% to 50% of free sequences

(which are not subjected to any constraints). The standard ratio would be 60% of strictly non

decremental sequences and 40% of free sequences.

9- From 9 up to 12 of only public sequences (the standard amount would be 11).

10- Each item should contain the minimum amount of terms which propagate a unique solution

(only one possible step-to-step pattern being involved), provided that the amount of the terms

belonging to the item itself is at least 7. Otherwise, the item has to list the first 7 terms of the final

sequence.

9

11- The psychologist/psychometrist who administers the test should take from 30 to 45 minutes to

properly explain the test rules to the candidates. At the beginning of the informative briefing, he

must deliver the full equipment package to each candidate.

12a Whenever one (or more) of the previous parameters is different from a given test to another

one, different norms must be used for the scoring.

12b (Reminder)- Last but not least, each blank term (indicated by a question mark) must be the

result of only one retro-analytical reasoning pattern.

An example of an IQ test for the high range, probably suitable for candidates with an IQ at or above

the 3σ cut-off [4] (e.g. students who constitute a valuable reservoir from which prestigious

universities around the world can draw resources), might be composed of a total of 35 (shuffled)

items, broken down as follows:

{

( )

( )

An appropriate time limit would be around 140-180 minutes.

It is important to point out that, considering “n” public sequences (IS), and “k” derivation steps, we

obtain ( ) total sequences. For example, if n=11 and k=3, we get 14641 possible sequences

(OS), even if some of them will not be useful for our purpose (to create valid items for the test).

Despite this being a little inconvenient, a large amount of sequences for a lot of different tests

(based on the same set of IS) will still be available.

Under the previous assumption, we have:

1 step 121 sequences (at most);

2 steps 1331 sequences (at most);

3 steps 14641 sequences (at most).

Starting from the given set of IS, the software produces one printout for each derivation step used

(i.e. a 3 steps OS creates 3 distinct printouts of different magnitude). The IS set (the input) is

changed automatically by the software itself, until the generated printout contains (at least) 6

distinct tests (whereas, under the assumptions above, the maximum quantity of different achievable

tests is equal ‒ using the “floor operator” ‒ to ⌊

⌋, where “n” represents the numerousness of the IS

considered by the test).

N.B.

It is not possible that the software will be unable to produce a minimum of 6 tests with no common

“useful” items amongst them.

In practice (under the constraints we have stated), a possible scenario (for one of the five most

difficult items of the test) could be as follows:

10

Public set of incremental sequences:

{

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

N.B.

In this case, we have chosen sequences from [http://en.wikipedia.org/wiki/Integer_sequence] but we

can also consider a(n):=7*k+3 (where k=0,1,2,3,…), or a(n):=a(n-1)+n (where the first term is equal to,

let us say, 4), etc.

Public set of free sequences:

{

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )

Thus, a possible item can be: 2, 11, 1, ???, 13, 13, 1, ...

The only acceptable solution is “13 ; [6,9,1,4]”, in fact, we have chosen the Ulam numbers as IS

(the mother sequence), the Decimal expansion of “e” (deleting the terms equal to zeros) rule for the

first step, the Golomb’s numbers rule for the second one and, at the end, the Practical numbers rule

for the third step of derivation.

4.1 General description of the software (by G. Morelli & M. Ripà):

how does it work?

ASG (Automatic Sequences Generator) is software created in order to automate the generation

process of the IQ test items.

The input of the program consists of a set of 9, 10, 11 or 12 input sequences (“IS”).

For reasons of simplicity, from here on we fix the amount of the IS we consider at 11.

11

The IS are provided through an ASCII sequential file.

In order to identify each single sequence, the set of IS is sorted according to the position (from 1 to

11) of each sequence within the input file.

The output of the program consists of an ASCII sequential file, which contains any sequence that

can be obtained through 1, 2 or 3 “steps of derivation” (see below for a detailed explanation)

applied to the IS included in the input file.

In each output sequence, each element is preceded by a number in brackets, which indicates the

position of the element within the OS.

The output file is sorted in alphabetical order (according to the ASCII code of each byte). Thanks to

this kind of disposition, a generic OS which occupies the position number N can be quickly

compared to the ones in the positions number N-1 and N+1: this lets us immediately identify which

items can actually be used in the test without any risk of having more than one possible solution.

N.B.

Each OS is “self-describing”. This means that each OS contains (at the end of the sequence) from 1

to 4 reserved numerical fields, which thus constitutes the “unique code of the sequence”.

This code describes the “story” of a sequence, identifying, step by step, the whole derivation

process used for creating that sequence. Each filing of the code is a number between 1 and 11,

because it represents the position of an IS within the set of IS (which is sorted, as previously said).

In details:

the first field indicates the “mother sequence”;

the second field indicates the IS used as “derivation rule” during the first step of derivation;

the third field (if present) indicates the IS used as “derivation rule” during the second step of

derivation;

the fourth field (if present) indicates the IS used as “derivation rule” during the third step of

derivation.

Observation. Of course, the fields of the code of each OS are not taken into account during the sort

of the output file.

4.2 An output example

The following text shows three consecutive output sequences, randomly selected from the OS

created by the software (in this context, also listing the whole set of IS is not relevant):

…………………………………………………………………..

…………………………………………………………………..

…………………………………………………………………..

12

Code: <9,1,2,8>

Sequence: (1)3,(2)9,(3)11,(4)16,(5)18,(6)26,(7)41,(8)201 ……..

Code: <9,8,2,8>

Sequence: (1)3,(2)9,(3)12,(4)11,(5)19,(6)36,(7)141,(8)304 ……

Code: <9,0,3,1>

Sequence: (1)3,(2)9,(3)86,(4)26,(5)12,(6)126,(7)65,(8)210 ……

…………………………………………………………………..

…………………………………………………………………..

…………………………………………………………………..

Note that, when the software lists the sequences in the output file, it also automatically specifies,

before each sequence, the unique-code of the sequence itself.

Considering the previous example, we can quickly choose an item to be used in the IQ test, using

the second sequence of the list:

item: 3,9,12,???,19,36,141

solution: blank item = 11; history-code = [9,8,2,8]

Observation. Again, we underline how we can immediately verify that the series adjacent to the

one we have chosen differs from the latter in at least one element (discarding the elements which

occupy the same position as the blank item); since the OS are sorted alphabetically, this provides us

with the certainty of the uniqueness of the solution.

4.3 Main flow

The main flow of the program includes three principal blocks:

receive (and check) the input sequences;

perform the Derivation Process;

print the output (the OS).

The core of the algorithm is the “Derivation Process”.

This process consists of 1, 2 or 3 steps of derivation.

For each step, the program uses a “pair of sequences”.

In each pair:

13

The first sequence is the series to which the step of derivation must be applied.

It is a “primary” series (that is, an IS), if it is used during the first step of derivation;

otherwise, it is a “derived” sequence (that is, it is the output of the previous step of

derivation).

The second sequence represents the “rule” which will be applied to the first sequence of

the same pair. The second sequence of a pair is always an IS.

In the program, the second sequence of a pair is considered as an “array of positions”, because it

contains the positions that the algorithm uses to search, within the first sequence of the same pair,

the values to be inserted into the output sequence generated in the current step of derivation.

According to this pattern, the first sequence of a pair represents an “array of values”: a subset of

these values will constitute the output sequence of the step of derivation.

So, for each step of derivation, the software uses two sequences: the first one is denoted by

“seqVal”, because it contains the “values” used by the algorithm of derivation, and the second one

is denoted by “seqPos”, because it is an array of “positions”.

A third sequence involved in the step of derivation is the output sequence, which is denoted by

“outSeq”.

All of the sequences are managed through arrays of integer. The whole set of the IS and the whole

set of the OS are represented as matrices (two-dimensional arrays) of integers.

During a step of derivation, we run through the seqPos, as follows: for each element of this

sequence, the software inserts into the seqOut an integer equal to the element of seqVal which

occupies (within the seqVal) the position number “I”, where “I” represents the value of the current

element of the seqPos.

If, during the cycle, we get a value in the seqPos which is higher than the cardinality (total number

of elements different from -1, not including the fields of the “unique code”) of the seqVal, the

software adds a fictitious element “-1” to the outSeq and the cycle ends. This fictitious element

represents an “impossible value”, because we do not manage negative integers, so it only serves to

indicate that the cardinality of the seqVal has been overtaken during the creation of the outSeq.

Once the cycle is completed, the software updates the unique-code of the series.

Obviously, the main flow of the program performs all the steps of derivations which can be applied

to the set of IS; the total number of OS is equal to the sum of 3 terms:

11^2 : total number of OS which can be obtained through a single step of derivation;

11^3 : total number of OS which can be obtained through 2 steps of derivation;

11^4 : total number of OS which can be obtained through 3 steps of derivation.

Therefore, the global number of OS is equal to 11^2 + 11^3 + 11^4 = 16093.

14

4.4 Technical data

Programming language:

Java (jdk 6.0).

Hardware requirements:

CPU: Intel or AMD dual core processor or higher;

Memory: at least 2 GB of RAM;

Operating System: any version of Windows or Linux (Java software is platform

independent).

Elapsed time (total time spent for a complete elaboration on a system with the minimum

hardware requirements):

Between 90 and 120 seconds.

5 Operative rules (guidelines for the psychometrist who oversees the

test)

The test session starts with the presentation/explanation of the test itself. It is essential that each

candidate receives the same information, which must be provided in the same way by the

psychometrist.

The guidelines that each psychometrist should respect are as follows:

0- The test rules explanation has to take a minimum of 30 minutes and a maximum of 45, plus

15 minutes for answering candidate’s questions (if any). He should not respond about the

rules that he has clearly explained and he cannot give other details, except the ones which

are included in this list.

1- At the beginning of the “explanation time”, before starting to talk about the test, the

psychometrist must deliver a pamphlet to each candidate. He must also hand to each

candidate a protocol sheet, a pencil with a rubber on the top and a pen. The final answers

must be written on the answer sheet using the pen, while the pencil can be used by the

candidate during the “thinking process”, together with the rubber and the calculation paper.

No pocket calculators can be brought into the test room. The candidate is not allowed to

15

introduce further material in the test room, that is, in addition to what will be given to him

by the psychometrist: a pencil, a rubber, a pen, a protocol sheet, the informative pamphlet

(brochure) and, at the right time, the test itself.

2- Inside the test room there should be a clock to let each candidate check the remaining

available time before the submission deadline. The psychometrist must notify all candidates

about the end of the test.

3- The psychometrist should clearly describe the method used to construct/derive one sequence

from another one, mentioning an example which uses FibonacciFibonacci sequence and

showing the sequence FibonacciFibonacciPrime-numbers.

4- The psychometrist must denote by “IS” the mother sequence and by “OS” the final outputs

printed on the exam. He has to refer to “IS” and “OS” during the whole explanation of the

test rules.

5- The psychometrist has to clarify that for each derivation step, the underlying derivation rule

may change, regardless of the rules used in the previous steps of the same item.

6- The psychometrist has to clearly underline how a step of derivation manages the cardinality

(total number of elements different from “-1”) of the sequence “seqVal” to be derived: if in

the “seqPos” (the sequence whose values represent the positions of the elements of seqVal

which will constitute the output sequence) we get a value which is higher than the

cardinality of seqVal, then we add a fictitious element “-1” to the output sequence and the

current step of derivation ends. This fictitious element, as previously noted, represents an

“impossible value”, because we do not manage negative integers, so it only serves to

indicate that the cardinality of the seqVal has been overtaken during the creation of the

outSeq.

7- The psychometrist must globally spend 10 minutes out of the total available time (30-45

minutes) to read out the pamphlet in front of the candidates.

6 Test rules pamphlet

Before starting to elucidate the test rules, the psychometrist must hand (to each candidate) a two

page long pamphlet, containing the following instructions:

1- The test contains 35 total items of different difficulty. The items are NOT arranged

according to their difficulty (they are randomly shuffled).

2- The maximum amount of steps of derivation is equal to 3. Therefore, if you find an item

solution using more than 3 derivation steps, it means that your attempt to answer that item is

definitely wrong.

3- The item subsets are as follows: 3 items1 derivation; 20 items2 derivations (5 using the

same rule plus 15 using two distinct rules); 12 items3 derivations (7 using two different

rules plus 5 based on three different rules).

16

4- Any term belonging to the sequences and used for the derivations is smaller than 1010

(from

1 to 9 digit numbers).

5- Each sequence contains an unlimited quantity of terms and each OS contains from 7 up to

70 elements (including the blank term “???”).

6- 6 out of 11 total sequences are strictly not “non decreasing” (each term is not greater than

the next one), while 5 sequences are free (not recursive positive integer sequences which do

not respect the rule n(i)≥n(i-1)).

7- Each OS contains the minimum quantity of terms which promulgate a unique solution (only

one possible step-to-step pattern being involved), provided that the amount of the terms

belonging to the item itself is at least equal to 7. Otherwise, the item has to list the first 7

terms of the final sequence. The only exception is represented by solving patterns which

involve more than 500 terms.

7b (reminder)- Every blank term (identified with three consecutive question marks) is the result

of only one retro-analytical reasoning pattern involving, at most, 500 terms for each

derivation step.

8- In order to help the candidate during the solving process, (but only with aspects which are in

no way related to any kind of intellectual ability) each term of the IS is preceded by a

progressive number put between brackets. The latter represents the position of the given

term inside the IS it belongs to. For example, taking as IS the Fibonacci’s sequence, we

have: (1)1,(2)1,(3)2,(4)3,(5)5,(6)8,(7)13,(8)21,(9)34,(10)55,(11)89,(12)144,(13)233,…

9- The answer provided by the candidate must contain the ??? value (the unknown element)

and a sequence of numbers which describes the pattern used to produce the given item. For

example, using 3 derivation steps, a possible solution (for a given item) could be: “???=58;

[10,5,3,5]”. If any of these four boxes are left blank, or contain wrong values, the item score

is zero. If an item is not based on 3 derivation steps, the candidate can leave blank the final

box/boxes of the array or put inside one “X” for each blank box (e.g. “???=198 ; [6,2,X,X]”

or simply “???=198 ; [6,2, , ]”).

10- There are no penalties (i.e. negative scores) for wrong answers, so trying to guess is an

advantage for the candidate. However, the criterion used in order to discriminate among

candidates who achieve the same raw score takes into account the number of answers

provided by each participant: the fewer answers a candidate provides, the higher his final

rank will be (inside the given raw score group).

7.1 Norming process

After the implementation, we are ready for the operative testing phase. We require at least 200

ascertained gifted candidates (IQ 130+, σ=15 on some standardized tests). It is not important if we

administer one specific test or a group of equivalent tests with the same parameter settings: the

norm will be unique and this psychometric tool will be ready to be used for screening highly gifted

children, searching for the mathematically talented [5], future chess grandmasters or excellent

scientists.

17

Following the approach previously introduced, it will be theoretically possible to create a kind of

extremely/profoundly gifted young children database as well. The ranking, obtained through the

performance achieved on this innovative generation of tests for the high range, would be able to

open up new frontiers, letting us offer many new opportunities.

It would be optimal to link these achievements to some prizes and/or other awards from academic

institutions or cultural organizations. They would be assured that their future students are (at least)

“highly gifted”, without any doubt about possible cheating.

Alternatively, the test could be used for ranking purposes [1], referring to the grants explained

above. With reference to particular fields such as mathematics (number theory, group theory, etc.),

cryptanalysis, geometry and professional chess, the selection method based on the raw score

achieved on one test of this kind would be a more reliable talent indicator, especially if compared to

an equivalent performance on the GMAT test or the GRE.

7.2 A faster option for screening candidates

Premised on the fact that no norm, by construction, can ever be stable over time (due to the Flynn

effect, a general mean fluctuation related to various population means, etc.) [6-12] and that any IQ

related performance is affected by many external factors, there are other interesting options which

can be explored in order to reach our goal. To facilitate a reduction in wasted time, with the aim of

obtaining a cheaper operational procedure, it will be possible to use a battery of two different

performance tests on a not-previously-tested population: a collective tool, plus an individual one. In

particular, under the assumption that we have a target group of candidates above the +3σ from the

expected mean, we could use the Raven’s Matrices [9] setting the cut-off at +2σ from the mean.

Candidates who pass the first (preliminary) test would be admitted to the second phase (taking one

test belonging to the test family previously described). In fact, Raven’s Matrices-based tests (which

are among the highest g-loaded psychometric tools available) are quite similar to the one presented

in this paper.

8 Conclusion

At this time, there is no test for assessing high IQ that can be entirely immune from the risk of

cheating: both supervised and high range tests are prone to this problem. On the other hand, a lot of

high IQ people ask for a tool which is as truthful and reliable as possible. In particular, gifted

students’ screening is very important in order to invest in the future from a meritocratic perspective,

gaining individual richness from the evaluation of youths’ talents. By sustaining individual

capabilities we contribute to psychological good health, but there is more: it is a real strategic

resource, directed towards social development and human progress. Investing in talent will bring

benefits not only to the economy, innovation and employment, but will also help social cohesion,

progress and competitiveness, promoting the growth of the Knowledge Society.

18

For this purpose, we have developed one method which opens the door to a new generation of IQ

tests for the high range and which is able to protect any candidate from the risk of cheating by

others. The first advantage, compared with “normal” standardized tests, is that, once it has been

explained, it is entirely automatized, and the test itself is different from candidate to candidate,

changing from one examination session to another one as well. Nevertheless, each version

maintains exactly the same difficulty as the others, even if the chosen “IQ target” we are interested

in can be changed (in a given range) by simply modifying some of the parameters listed in section

3.

“Classical” numerical tests are the ones most commonly affected by cheating (it is very simple to

exchange information on a given item via the internet), while our family is absolutely secure. Our

tests are numerical, only involving integer sequences, but they are mainly focused on pattern

recognition and fast reasoning/deep analysis: they are similar to the best g-loaded tests, such as

Raven’s Matrices. For this reason, it is possible to adopt the RPM as a preliminary test to make an

early screening of the candidates, putting the IQ cut-off at +2σ from the mean.

Unlike other supervised tests, the ENNDT (Equally Normed Numerical Derivation Tests) are

specifically designed to investigate very high IQs (above the 3σ level).

In theory, a person can engage in them more than once, because his performance is not vitiated by

the training effect facilitated by multiple attempts. This could also help monitor the development of

very high IQs with increasing age.

Appendix

The derivation process illustrated, when applied to sequences constructed by the decimal expansion

of an irrational number, always produces, after a few iterations at the most, a very small quantity of

sequences which repeat cyclically.

In detail, we can analyze what happens considering three of the most famous irrationals: the number

of Euler (Napier), pi and √ [10].

Considering the decimal expansion of “e”, we have that (for n∈ℕ0), after 2n+2 iterations (steps), the

recurring string is as follows: 2,7,1,8,2,8,1,8,2,8,8,1,1,8,1,2,7,1,7,8,2,8,7,8,7,1,7,1,2,8,8,2,8,1,7,7,…

In fact, the cycles of the terms of the original sequence (A002193 of the OEIS) are:

1⤇2⤇1⤇...

2⤇7⤇2⤇...

3⤇1⤇2⤇1⤇...

4⤇8⤇8⤇...

5⤇2⤇7⤇2⤇...

6⤇8⤇8⤇...

7⤇1⤇7⤇...

8⤇8⤇...

9⤇2⤇7⤇2⤇...

0⤇//

19

N.B.

Remember that the terms which are “zero” are removed (by covenant) from the first step of the

derivation process.

Referring once more to the decimal expansion of “e”, we have that (for n∈ℕ0), after 2n+1

derivations (steps), and the recurring string is similar to the sequence A119506 of the OEIS (the

only dissimilarity is represented by the different interpretation given to the elements “0” in the

original sequence ‒ with the related transformation cycle).

Observing the decimal expansion of “pi”, we have that (for n∈ℕ0) after 2n+2 derivations (steps), the

recurring string is: 3,1,4,1,5,5,4,5,5,3,5,5,5,3,5,3,4,3,5,4,5,4,3,3,5,…

In fact, the cycles of the terms of the original sequence (A000796 of the OEIS) are:

1⤇3⤇1⤇...

2⤇1⤇4⤇1⤇...

3⤇4⤇3⤇...

4⤇1⤇4⤇...

5⤇5⤇...

6⤇9⤇5⤇5⤇...

7⤇2⤇3⤇4⤇3

8⤇6⤇5⤇5⤇…

9⤇5⤇5⤇...

0⤇//

Again, with reference to the decimal expansion of “pi”, it follows that (for n∈ℕ0), after 2n+3

successive derivations, the (unique) recurring string is as follows:

4,3,1,3,5,5,1,5,5,4,5,5,5,4,5,4,1,4,5,1,5,1,4,4,5,…

(while, after the first step, we have 4,3,1,3,5,5,1,9,5,4,5,6,5,2,5,4,1,4,6,1,9,1,9,1,4,4,6,…).

Finally, analyzing the decimal development of √ , it follows that (for n∈ℕ0) after n+2 derivations

(steps), the (unique) recurring string is:

1,4,1,4,4,1,1,4,1,4,1,1,1,1,4,4,4,4,1,1,4,4,1,4,4,4,1,1,1,4,1,4,4,1,1,1,1,…

(while, after the first step, we have 1,4,1,4,4,1,1,2,1,4,1,3,1,6,2,4,5,5,1,1,5,5,3,4,4,4,6,1,6,5,

3,5,5,9,6,9,3,…).

Thus, the cycles of the terms of the original sequence (A002193 of the OEIS) are:

1⤇1⤇...

2⤇4⤇4⤇...

3⤇1⤇1⤇...

4⤇4⤇...

5⤇2⤇4⤇4⤇...

6⤇1⤇1⤇...

7⤇3⤇1⤇1⤇...

8⤇5⤇4⤇4⤇...

9⤇6⤇1⤇1⤇...

0⤇//

20

Acknowledgements

The authors sincerely thank Dr. Evangelos Katsioulis, Dr. Manahel Thabet and Graham Powell for

their invaluable support.

References

[1] J. Betts, World Genius Directory (2012). http://www.psiq.org/

[2] P. Cooijmans, Recommendations for conducting high-range intelligence tests (2010).

http://www.paulcooijmans.com/intelligence/recommend.html

[3] P.Cooijmans, IQ development with age modeled (08/2010).

http://www.paulcooijmans.com/intelligence/iq_development_with_age_modelled.html

[4] R. de la Jara, IQ percentile and rarity chart. http://www.iqcomparisonsite.com/IQtable.aspx

[5] Educational Studies in Mathematics 17, pp. 243-259, Identification and fostering of

mathematically gifted students, D. Reidel Publishing Company (1986).

[6] R. D. Fuerle, A Possible Explanation for the Flynn Effect (01/2008).

http://majorityrights.com/weblog/comments/a_possible_explanation_for_the_flynn_effect

[7] O. Janko, The retrograde analysis corner (2010). http://www.janko.at/Retros/index.htm

[8] P. L. Miranda, High range IQ tests (2007-2012). http://webs.ono.com/iqtests/

[9] M. Ripà, Conference proceedings, 12th Asia-Pacific Conference on Giftedness, Dubai

(17/08/2012). http://www.scribd.com/doc/104649946/Identifying-Gifted-Children-and-Dyslexia-

Early-Diagnosis-Conference-proceedings-12th-Asia-Pacific-Conference-on-Giftedness-July-17-

2012

[10] M. Ripà, pp. 36-40, Congetture su interrogativi inediti: tra speculazioni, voli pindarici e

riflessioni spicciole, Narcissus Self Publishing (06/2012).

[11] 60 Minutes: Smarty Pants, Air date: 21/05/2010 (7:30). http://ondemand.tv3.co.nz/60-Minutes-

60-Minutes-SmartyPants/tabid/59/articleID/2493/MCat/22/Default.aspx

[12] J. Walker, Global IQ: 1950-2050, Fourmilab, (04/2004).

http://www.fourmilab.ch/documents/IQ/1950-2050/